Applied Engineering Mathematics

Applied Engineering Mathematics

Xin-She Yang

University of Cambridge, Cambridge, United Kingdom

7 l\!Ieadow Walk, Great Abington, Cambridge CBl 6AZ, UK http:/

fwww

.cisp-publishing.com

First Published 2007

@Cambridge International Science Publishing @Xin-She Yang

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All rights reserved. No part of this publication may be repro-duced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any infornla-tion storage and retrieval system, without permission in writing from the copyright holder.

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ISBN 978-1-904602-56-9

Cover design by Terry Callanan

Preface

Engineering mathematics including numerical methods and application is the essential part of key problem-solving skills for engineers and scientists. Modern engineering design and pro-cess modelling require both mathematical analysis and com-puter simulations. Vast literature exists on engineering math-ematics, mathen1atical modelling and numerical methods. The topics in engineering mathematics are very diverse and the syl-labus of mathematics itself is evolving. Therefore, there is a decision to select the topics and limit the number of chapters so that the book remains concise and yet comprehensive enough to include all the important mathematical methods and popu-lar numerical methods.

This book endeavors to strike a balance between mathemat-ical and numermathemat-ical coverage of a wide range of mathematmathemat-ical methods and numerical techniques. It strives to provide an introduction, especially for undergraduates and graduates, to engineering mathematics and its applications. Topics include advanced calculus, ordinary differential equations, partial dif-ferential equations, vector and tensor analysis, calculus of varia-tions, integral equavaria-tions, the finite difference method, the finite volume method, the finite element method, reaction-diffusion system, and probability and statistics. The book also emph~

sizes the application of important mathematical methods with dozens of worked examples. The applied topics include elastic-ity, harmonic motion, chaos, kinematics, pattern formation and hypothesis testing. The book can serve as a textbook in en-gineering mathematics, mathematical modelling, and scientific computing.

Xin-She Yang Cambridge, 2007

First and foremost, I would like to thank my mentors, tutors and colleagues: Prof. A C Fowler and Prof. C J :Mcdiarmid at Oxford University, Dr J l\II Lees and Dr C T Morley at Cambridge University, Prof. A C :rvlclntosh, Prof. J Brindley, Prof. R W Lewis, Prof. D T Gethin, and Prof. Andre Revil for their help and encouragement. I also thank Dr G. Parks, Dr T. Love, Dr S. Guest, Dr K. Seffen, and many colleagues for their inspiration. I thank many of my students, especially Hugo Whittle and Charles Pearson, at Cambridge University who have indirectly tried some parts of this book and gave their valuable suggestions.

I also would like to thank my publisher, Dr Victor Riecan-sky, for his kind help and professional editing.

Last but not least, I thank my wife, Helen, and son, Young, for their help and support.

About the Author

Xin-She Yang received his D.Phil in applied n1athematics from the University of Oxford. He is currently a research fellow at the University of Cambridge. Dr Yang has published exten-sively in international journals, book chapters, and conference proceedings. His research interests include asymptotic anal-ysis, bioinspired algorithms, combustion, computational engi-neering, engineering optimization, solar eclipses, scientific pro-gramming and pattern formation. He is also the author of a book entitled: "An Introduction to Computational Engineering with Jvlatlab", published in 2006 by Cambridge International Science Publishing Ltd.

Contents

1 Calculus 1 1.1 Differentiations

. . .

1 1.1.1 Definition . . . 1 1.1.2 Differentiation Rules 2 1.1.3 In1plicit Differentiation . 4 1.2 Integrations . . . 5 1.2.1 Definition

...

5 1.2.2 Integration by Parts

..

6

1.2.3 Taylor Series and Power Series 8

1.3 Partial Differentiation

...

9 1.3.1 Partial Differentiation

....

9 1.3.2 Differentiation of an Integral 12 1.4 Multiple Integrals . . . 12 1.4.1 l\1ultiple Integrals 12 1.4.2 Jacobian . . . 13

1.5 Some Special Integrals . . 16

1.5.1 Asymptotic Series 17 1.5.2 Gaussian Integrals 18 1.5.3 Error Functions . . 20 1.5.4 Gamma Functions 22 1.5.5 Bessel Functions 24 2 Vector Analysis 27 2.1 Vectors

...

.

. .

.

. . . .

27

2.1.2 Cross Product 2.1.3 Vector Triple . 2.2 Vector Algebra . . . . 2.2.1 Differentiation of Vectors 2.2.2 Kinematics . . . . 2.2.3 Line Integral . . . . 2.2.4 Three Basic Operators .. 2.2.5 Son1e Important Theorems 2.3 Applications . . . . 2.3.1 Conservation of 1vlass 2.3.2 Saturn's Rings 3 Matrix Algebra 3.1 :rviatrix . . . . 3.2 Determinant. 3.3 Inverse . . . . 3.4 :rviatrix Exponential.

3.5 Hermitian and Quadratic Forms 3.6 Solution of linear systems 4 Complex Variables

4.1 Complex Numbers and Functions . 4.2 Hyperbolic Functions .

4.3 Analytic Functions 4.4 Complex Integrals . .

5 Ordinary Differential Equations 5.1 Introduction . . . . .

5.2 First Order ODEs . . . 5.2.1 Linear ODEs . . 5.2.2 Nonlinear ODEs 5.3 Higher Order ODEs ..

5.3.1 General Solution 5.3.2 Differential Operator . 5.4 Linear System . . . . 5.5 Sturm-Liouville Equation 30 31 32 32 33

37

38 40 41 41 42 47

47

49 50 52 53 56 61 61 65 67 70 77

77

78 78 80 81 81 84 85 86

CONTENTS

5.5.1 Bessel Equation . . . . 5.5.2 Euler Buckling . . . . 5.5.3 Nonlinear Second-Order ODEs 6 Recurrence Equations

6.1 Linear Difference Equations 6.2 Chaos and Dynamical Systems

6.2.1 Bifurcations and Chaos 6.2.2 Dynamic Reconstruction. 6.2.3 Lorenz Attractor .. 6.3 Self-similarity and Fractals . . . 7 Vibration and Harmonic Motion

7.1 Undamped Forced Oscillations 7.2 Damped Forced Oscillations . 7.3 Normal Ivlodes . . . . 7.4 Small Amplitude Oscillations 8 Integral Transforms 8.1 Fourier Transform 8.1.1 Fourier Series . . 8.1.2 Fourier Integral . 8.1.3 Fourier Transform 8.2 Laplace Transforms. 8.3 ~avelet . . . . 9 Partial Differential Equations

9.1 First Order PDE 9.2 Classification 9.3 Classic PDEs . .

10 Techniques for Solving PDEs 10.1 Separation of Variables . 10.2 Transform l\1ethods . . . . 10.3 Similarity Solution . . . . 10.4 Travelling ~ave Solution .

CONTENTS

88

90 91 95 95

98

99

. 102 . 103 . 105 109 . 109 . 112 . 116 . 119

125

. 126 . 126 . 128 . . 129 . 131 . 134

137

. 138 . 139 . 139 141 . 141 . 143 . 145 . 147

10.5 Green's Function 10.6 Hybrid Method . 11 Integral Equations 11.1 Calculus of Variations . . . . 11.1.1 Curvature . . . . 11.1.2 Euler-Lagrange Equation 11.1.3 Variations with Constraints 11.1.4 Variations for l\1ultiple Variables 11.2 Integral Equations . . . .

11.2.1 Linear Integral Equations 11.3 Solution of Integral Equations .

11.3.1 Separable Kernels . . 11.3.2 Displacement Kernels 11.3.3 Volterra Equation 12 Tensor Analysis 12.1 Notations .. 12.2 Tensors . . . 12.3 Tensor Analysis . 13 Elasticity

13.1 Hooke's Law and Elasticity 13.2 l\1axwell's Reciprocal Theorem 13.3 Equations of l\1otion . . . . . 13.4 Airy Stress Functions . . . . 13.5 Euler-Bernoulli Beam Theory

. 148 . 149 153 . 153 . 153 . 154 . 160 . 165 . 167 . 167 . 169 . 169 . 170 . 170 173 . 173 . 174 . 175 181 . 181 . 185 . 189 . 192 . 196 14 Mathematical Models 201 14.1 Classic l\1odels . . . . 201 14.1.1 Laplace's and Poisson's Equation . . 202

14.1.2 Parabolic Equation . . 202

14.1.3 Wave Equation . . . . 203 14.2 Other PDEs . . . . 203 14.2.1 Elastic Wave Equation . . 203 14.2.2 l\tlaxwell's Equations . 204

CONTENTS 14.2.3 Reaction-Diffusion Equation. 14.2.4 Fokker-Plank Equation 14.2.5 Black-Scholes Equation . 14.2.6 Schrodinger Equation .. 14.2.7 Navier-Stokes Equations . 14.2.8 Sine-Gordon Equation CONTENTS . 204 . 205 . 205 . 206 . 206 . 207

15 Finite Difference Method 209

15.1 Integration of ODEs . . . 209

15.1.1 Euler Scheme . . . 210

15.1.2 Leap-Frog Jviethod . 212

15.1.3 Runge-Kutta Jviethod . 213

15.2 Hyperbolic Equations . . . . . 213 15.2.1 First-Order Hyperbolic Equation . 214 15.2.2 Second-Order Wave Equation . 215

15.3 Parabolic Equation . . 216

15.4 Elliptical Equation . . 218

16 Finite Volume Method 221

16.1 Introduction . . . . 221

16.2 Elliptic Equations . . 222

16.3 Parabolic Equations . 223

16.4 Hyperbolic Equations . 224

17 Finite Element Method 227

17.1 Concept of Elements . . . . 228 17.1.1 Simple Spring Systems . . . 228 17.1.2 Bar and Beam Elements . . 232

17.2 Finite Element Formulation . 235

17.2.1 Weak Formulation . 235

17.2.2 Galerkin Jviethod . 236

17.2.3 Shape Functions . . . 237

17.3 Elasticity . . . . 239 17.3.1 Plane Stress and Plane Strain . . 239

17.3.2 Implementation . . 242

17.4.1 Basic Formulation . . . . 244 17 .4.2 Element-by-Element Assembly . . . . 246 17.4.3 Application of Boundary Conditions . 248

17.5 Time-Dependent Problems . . . 251

17.5.1 The Time Dimension. . . . 251 17.5.2 Time-Stepping . . . . 253 17.5.3 1-D Transient Heat Transfer . . 253

17.5.4 Wave Equation .. . 254

18 Reaction Diffusion System 257

18.1 Heat Conduction Equation . 257

18.1.1 Fundamental Solutions . . 257 18.2 Nonlinear Equations . . . . 259

18.2.1 Travelling Wave . . . 259

18.2.2 Pattern Formation . . 260

18.3 Reaction-Diffusion System . . 263

19 Probability and Statistics 267

19.1 Probability . . . . 267 19.1.1 Randomness and Probability . 267 19.1.2 Conditional Probability . . . . 275 19.1.3 Random Variables and Ivloments . 277 19.1.4 Binomial and Poisson Distributions. . 281 19.1.5 Gaussian Distribution . . . . . 283 19.1.6 Other Distributions . . . . 286 19.1.7 The Central Limit Theorem . . 287 19.2 Statistics . . . . 289 19.2.1 Sample Ivlean and Variance . 290 19.2.2 Iviethod of Least Squares . 292

19.2.3 Hypothesis Testing . . 297

A Mathematical Formulas 311

A.1 Differentiations and Integrations . 311

A.2 Vectors and Matrices . . 312

A.3 Asymptotics . 314

Chapter 1

Calculus

The prelin1inary requirements for this book are the pre-calculus foundation mathematics. We assume that the readers are fa-miliar with these preliminaries, and readers can refer to any book that is dedicated to these topics. Therefore, we will only review some of the basic concepts of differentiation and inte-gration.

1.1 Differentiations

1.1.1 Definition

For a known function or a curve y =

f (

x) as shown in Figure 1.1, the slope or the gradient of the curve at the point P( x, y) is defined as

dy

=

df(x)

=

!'(x) = lim f(x

+

~x)- f(x), (1.₁₎

dx dx .dx--+0 ~x

on the condition that there exists such a limit at P.

This gradient or limit is the first derivative of the function

f (

x) at P. If the limit does not exist at a point P, then we say that the function is non-differentiable at P. By conven-tion, the limit of the infinitesimal change ~x is denoted as the differential dx. Thus, the above definition can also be written

as

dl(x) 1

dy

=

dl

=

~dx

=I

(x)dx, (1.2) which can be used to calculate the change indy caused by the small change of dx. The primed notation 1 _{and standard} nota-tion d~ _{can be used interchangeably, and the choice is purely} out of convenience.

Figure 1.1: Gradient of a curve

The second derivative of l(x) is defined as the gradient of

l

1_(x), or

d2y

=I"( )

= dl(x)

dx2- x dx · (1.3)

The higher derivatives can be defined in a similar manner. Thus, d3y

=I"'( ) =

dl"(x) dx3 - x dx ' lf'ly dl(n-l)

... , - =

l(n) = . dxn dx (1.4)

1.1. 2 Differentiation Rules

If a more complicated function

I (

x) can be written as a prod-uct of two simpler functions u(x) and v(x), we can derive a differentiation rule using the definition from the first

princi-Calculus 1.1 Differentiations

pies. Using

f(x

+

~x) - f(x) u(x

+

~x)v(x

+

~x) - u(x)v(x)

=

and subtracting and adding -u(x

+

~x)v(x)

+

u(x

+

~x)v(x)

[= 0] terms, we have

df d[u(x)v(x)] dx = dx

I1m . [ ( ux+~x A )v(x

+

~x)-v(x) +vx ( )u(x

+

~x)-~ u(x)]

~~ ~ X

dv du

= u(x) dx

+

dx v(x), (1.5) which can be written in a contract form using primed notations

J'(x)

=

(uv)'

=

u'v

+

uv'. (1.6) If we differentiate this equation again and again, we can gen-eralize this rule, we finally get the Leibnitz's Theorem for dif-ferentiations

+ ... +

uv(n), _(1.7)

where the coefficients are the same as the binomial coefficients

nc

= (

n)

=

nl

r - r rl(n-r)!' (1.8)

If a function f(x) [for example, f(x) = exn] can be written

as a function of another function g(x), or f(x)

=

f[g(x)] [for example, f(x)

=

eg(x) and g(x)

=

xn], then we have

! '( ) -

X - I' l i D - -

~~ ~g

which leads to the following chain rule f1_{(x) = df dg,} dgdx or {f[g(x)]}1 = J'[g(x)] · g1(x). In our example, we have

f' (

x) = (ex")' = ex" nxn -l.

(1.10)

(1.11)

If one use 1/v instead of v in the equation (1.6) and (1/v)' =

-v1 _fv2_{, we have the following differentiation rule for quotients:}

I I

(!)I = u v - uv .

v v2 (1.12)

0 Example 1.1: The deriva.tive of f(x) = sin(x)e-cos(x) can be obtained using the combination of the above differentiation rules.

f'(x) = [sin(x))'e-cos(x)

+

sin(x)[e-cos(x))' = cos(x)e-cos(x)

+

sin(x)e- cos(x)[- cos(x))'

= cos(x )e-cos(x)

+

sin2(x )e-cos(x).

The derivatives of various functions are listed in Table 1.1.

1.1.3 Implicit Differentiation

The above differentiation rules still apply in the case when there is no simple explicit function form y =

f (

x) as a function of

x only. For example, y

+

sin(x) exp(y) = 0. In this case, we can differentiate the equation tern1 by tern1 with respect to x so that we can obtain the derivative dyfdx which is in general a function of both x and y.

0 Example 1. 2: Find the derivative ::; if y2

+

sin( x )eY = cos( x). Differentiating term by term with respect to x, we have

2y

~~

+

cos(x)eY

+

sin(x)eY:

= -

sin(x),

dy cos(x)eY

+

sin(x)

dx - 2y

+

sin(x)eY ·

Calculus 1.2 Integrations

Table 1.1: First Derivatives

f(x) f'(x) xn nxn-1 ex ex ax(a

> 0)

axlna lnx ! _X logax _xlna 1 sinx cosx cosx -sinx tanx sec2 x sin-1 _x 1 ~ cos-1 _x 1 - v'1-x2 tan- 1 x 1 1+x2 sinhx coshx coshx sinhx

1.2 Integrations

1.2.1 Definition

Integration can be viewed as the inverse of differentiation. The integration F( x) of a function

f (

x) satisfies

d~~x)

= f(x), (1.13)

or

F(x)

=

1x

!(~)~,

xo

(1.14) where

f (

x) is called the integrand, and the integration starts from xo (arbitrary) to x. In order to avoid any potential confu-sion, it is conventional to use a dummy variable (say, ~) in the integrand. As we know, the geometrical meaning of the first derivative is the gradient of the function

f (

x) at a point P, the

geometrical representation of an integral

J:

f(~)d~ (with lower integration limit a and upper integration limit b) is the area under the curve f(x) enclosed by x-a.xis in the region x E [a, b].

In this case, the integral is called a definite integral as the limits are given. For the definite integral, we have

fb

f(x)dx

=

1b

f(x)dx

-1a

f(x)dx

=

F(b)- F(a). (1.15)

la xo xo

The difference F (b) - F (a) is often written in a compact form Fl~

=

F(b)- F(a). As F'(x)

=

f(x), we can also write the above equation as

l

f(x)dx =

l

F'(x)dx = F(b) - F(a). (1.16) Since the lower limit x0 is arbitrary, the change or shift of the lower limit will lead to an arbitrary constant c. When the lower limit is not explicitly given, the integral is called an indefinite integral

I

f(x)dx = F(x)

+

c, (1.17) where cis the constant of integration.

The integrals of some of the con1mon functions are listed in Table 1.2.

1.2.2 Integration by Parts

From the differentiation rule ( uv )' = uv'

+

u' v, we have

uv' = (uv)'- u'v. (1.18) Integrating both sides, we have

u-dx = uv - -vdx

I

dv ldu

dx dx ' (1.19)

in the indefinite form. It can also be written in the definite form as

1

b

u-d dx dv = [uv]

lb 1b

+

v-d dx. du

Calculus 1.2 Integrations Table 1.2: Integrals f(x)

f

f(x) xn(n

tf

-1) xn n+1 1 _lnlxl X ex ex sinx -cosx cosx sinx 1 _{1 tan-}1 _i!. a2tx2 a a 1 In a+x a2-x2 2a a-x 1 _..!..In~ x2-a2 2a x+a 1 _sin-1 _~ v'a1-x1 a 1 _{ln(x + v'x2 + a2)} ~ [or sinh-1 ~] 1 _ln(x

₊

_{v'x2 - a2)} Jx2-a2 (or cosh-1 ~] sinhx coshx coshx sinhx tanhx lncoshx

The integration by parts is a very powerful method for evalu-ating integrals. 1\.fany complicated integrands can be rewritten as a product of two simpler functions so that their integrals can easily obtained using integration by parts.

0 Example 1.9: The integral of I=

J

x lnx dx can be obtained by setting v' = x and u = lnx. Hence, v = x₂

2

and u' = ~· \-Ve now have I=

J

xlnxdx = - - -x --dx 2 lnx

J

x2 1 2 2 X 0

Other important methods of integration include the sub-stitution and reduction methods. Readers can refer any book that is dedicated to advanced calculus.

1.2.3 Taylor Series and Power Series

From

l

f(x)dx

=

F(b) - F(a),

and

1:£

=

F'

=

f(x), we have

1

xo+h /'(x)dx

=

f(xo +h)- f(xo),

xo

which means that

1

xo+h f(xo +h)

=

f(xo)

+

f'(x)dx. xo (1.21) (1.22) (1.23) If h is not too large or

f' (

x) does not vary dramatically, we can approximate the integral as

1

xo f'(x)dx ~ !'(xo)h.

xo (1.24)

Thus, we have the first-order approximation to

f (

xo

+

h)

f(xo +h)~ f(xo)

+

hf'(xo). (1.25) This is equivalent to say, any change from xo to xo+h is approx-imated by a linear term hf'(xo). If we repeat the procedure for

f'(x), we have

J'(xo +h)~ !'(xo)

+

hf"(xo), (1.26) which is a better approximation than f'(xo +h) ~ f'(x0 ). Fol-lowing the same procedure for higher order derivatives. we can reach the n-th order approximation

Calculus 1.3 Partial Differentiation

(1.27) where Rn+t (h) is the error of this approximation and the no-tation means that the error is about the same order as n

+

1-th term in the series. This is the well-known Taylor theorem and it has many applications. In deriving this formula, we have im-plicitly assumed that all the derivatives l'(x), l"(x), ... , l(n)(x) exist. In almost all the applications we meet, this is indeed the case. For example, sin( x) and ex, all the orders of the deriva-tives exist. If we continue the process to infinity, we then reach the infinite power series and the error limn-oo Rn+ 1 ---+ 0 if the

series converges. The end results are the Maclaurin series. For example, and

x3

xs

sin x = x - -

+ - - ... ,

(x E 'R), 3! 5! x2 x4 cosx = 1 - -

+-- ... ,

(x E 'R), 2! 4!

1.3 Partial Differentiation

1.3.1 Partial Differentiation

(1.28) (1.29) (1.30)

The differentiation defined above is for function

I (

x) which has only one independent variable x, and the gradient will generally depend on the location x. For functions

I (

x, y) of two variables

x andy, their gradient will depend on both x andy in general. In addition, the gradient or rate of change will also depend on the direction (along x-axis or y-axis or any other directions). For example, the function l(x, y)

=

xy shown in Figure 1.2 has different gradients at ( 0, 0) along x-axis and y-axis. The

1 -1

Figure 1.2: Variation of f(x, y) = xy.

gradients along the positive x- and y- directions are called the partial derivatives respect to x and y, respectively. They are denoted as

U

and

%,

respectively.

The partial derivative of

f (

x, y) with respect to x can be calculated assuming that y =constant. Thus, we have

af(x,y)

=

f

=

8!1

8x

-

x -

8x

Y = lim f(x

+

~x, y)- f(x, y) ~X .6.x-O,y=const Similarly, we have af(x,y)

=

f

=

8!1

ay - y-ayx lim .6.y-O ,x=const f(x, y

+

~y) - f(x, y) ~y (1.32) (1.33) The notation

fxly emphasizes the fact that y is held constant.

The subscript notation fx (or /y) emphasizes the derivative is carried out with respect to x (or y). :Mathematicians like to

Calculus 1.3 Partial Differentiation

use the subscript forms as they are simpler notations and can be easily generalized. For example,

(1.34) Since flxfly = flyflx, we have fxy = !yx·

0 Example 1.4: The first partial derivatives of f(x, y) = xy

+

sin(x)e-Y are

8!

fx = ax = y

+

Cos(x)e-Y,

The second partial derivative of f(x, y) is

fxx = -sin(x)e-Y,

and

fxy = fyx = 1-cos(x)e-Y.

0

For any small change Llf = f(x+flx, y+fly)- f(x, y) due to flx and fly, the total infinitesimal change df can be written as

(1.35) If x and y are functions of another independent variable ~, then the above equation leads to the following chain rule

df af dx af dy

~

=

8xd~

+

8y~'

(1.36)

which is very useful in calculating the derivatives in parametric form or for change of variables. If a complicated function

f (

x) can be written in terms of simpler functions u and v so that f(x) = g(x, u, v) where u(x) and v(x) are known functions of

x, then we have the generalized chain rule dg ag ag du ag dv

The extension to functions of more than two variables is straightforward. For a function p(x, y, z, t) such as the pressure in a fluid, we have the total differential as

df = op dt

+

op dx

+

op dy

+

8P dz. (1.38)

8t 8x oy 8z

1.3.2 Differentiation of an Integral

When differentiating an integral

<11(x)

=

l

<P(x, y)dy, (1.39) with fixed integration limits a and b, we have

84>(x) =

[b

8</J(x, y) dy.

OX

Ja

OX (1.40)

When differentiating the integrals with the limits being func-tions of x,

I(x)

=

ru(x) 1/J(x, T)dT

=

w[x, u(x)] - w[x, v(x)], lv(x)

the following formula is useful:

dl 1u(x) o'l/J du dv

-d =

-8 dT

+

[1/J(x, u(x))-d -1/J(x, v(x))-d ].

X v(x) X X X

This formula can be derived using the chain rule dl ol ol du 8 I dv

- = - + - - + - -

_{dx 8x ou dx ov dx'}

where ~

=

1/J(x, u(x)) and

§f

=

-1/J(x, v(x)).

1.4 Multiple Integrals

1.4.1 Multiple Integrals

(1.41)

(1.42)

(1.43)

As the integration of a function

f (

x) corresponds to the area enclosed under the function between integration limits, this

Calculus 1.4 A1ultiple Integrals

can extend to the double integral and multiple integrals. For a function f(x, y), the double integral is defined as

F

=In

f(x, y)dA, (1.44) where dA is the infinitesin1al element of the area, and

n

is the region for integration. The simplest form of dA is dA

=

dxdy

in Cartesian coordinates. In order to emphasize the double integral in this case, the integral is often written as

I =

J

In

f(x, y)dxdy. (1.45) 0 Example 1. 5: The area moment of inertia of a thin rectangular plate, with the length 2a and the width 2b, is defined by

I=

J

In

y2dS

=

J

In

y2dxdy.

The plate can be divided into four equal parts, and we have

0

1.4.2 Jacobian

Sometimes it is necessary to change variables when evaluating an integral. For a simple one-dimensional integral, the change of variables from x to a new variable v (say) leads to x = x(v).

This is relatively sin1ple as dv

=

~~dv, and we have

1

xb

1b

dv

f(x)dx = f(x(v))----d dv,

Xa a X (1.46)

where the integration limits change so that x(a) Xa and

x(b) = Xb. Here the extra factor dxjdv in the integrand is referred to as the Jacobian.

For a double integral, it is more complicated. Assuming x = x(~, 17), y = y(~, 17), we have

jj

f(x, y)dxdy =

j j

!(~1J)IJid~d1J,

(1.47) where J is the Jacobian. That is

J

=

a(x, y) a(~,1J)

I

{)x {)x

I I

{)x §JL

I

- ~ "lJ7i - 0~ 0~ - §JL §JL - {)x {JiJ • 0~ 0~ ~ "lJ7i (1.48) The notation

a(

x, y) /a(~, 1J) is just a useful shorthand. This equivalent to say that the change of the infinitesimal area dA = dxdy becomes

d d

=

I a(x, y) 1,/Cd

=

I ax ay - ax ay 1,/Cd

X y a(~, 1J) ~ 1J a~ a1] a1] a~ ~ 1]. (1.49) 0 Example 1. 6: \Vhen transforming from ( x, y) to polar coordi-nates (r, 8), we have the following relationships

x = rcos8, y = rsin8.

Thus, the Jacobian is

J _ a( X, y) _ ax ay ax ay

- a(r,8) - ar a8- ae ar

= cos 8 x r cos() - ( -r sin 8) x sin 8 = r[cos2 ₈

₊

_sin2 _{8) = r.}

Thus, an integral in (x, y) will be transformed into

/!

(j>(x,y)dxdy =

J

J

4>(rcos8,rsin8)rdrd().

0

In a similar fashion, the change of variables in triple inte-grals gives

Calculus 1.4 !v!ultiple Integrals and Ox Oy 0::: J

=

{)(X, y, Z)

=

~ Ox

~

~ 0::: _(1.51) iJ1] F F - {)(~, 1J, () _{Ox 0~ 0~} 0< 0<

m:

For cylindrical polar coordinates ( r, (jJ, z) as shown in Figure 1.3, we have

x

=

rcos(jJ, y = rsin¢, The Jacobian is therefore

cos 4> sin¢ 0 J

=

0( X, y, Z)

=

8(r,(jJ,z) -rsin¢ rcos¢ 0 0 0 1

z

Z =Z. =r.

Figure 1.3: Cylindrical polar coordinates.

(1.52)

(1.53)

For spherical polar coordinates (r, 8, 4>) as shown in Figure 1.4, where 8 is the zenithal angle between the z-axis and the position vector r, and 4> is the azimuthal angle, we have

x

=

rsin8cos¢, y

=

r sin 8 sin¢, z = r cos 8. (1.54) Therefore, the Jacobian is

sin 8 cos 4> sin 8 sin 4J cos 8

J = r cos 8 cos 4J r cos 8 sin 4> -r sin 8 = r2 sin 8. (1.55)

z

Figure 1.4: Spherical polar coordinates.

Thus, the volu1ne element change in the spherical systen1 is

dxdydz

=

r2 sin OdrdOdc/>. (1.56)

0 Example 1. 7: The volume of a solid ball with a radius R is defined as

V=

!!In

dV.

Since the infinitesimal volume element dV = r2 _{sin( O)drdOd¢ in}

spher-ical coordinates r ~ 0, 0 ~ () ~ 1r and 0 ~ ¢ ~ 21r, the ball can be divided into two equal parts so that

{R {rr/2 {21r V = 2

Jo {}

0 sin0[}0 d¢]dO}dr {R {rr/2 = 2

Jo

{27r

Jo

sin(O)dO}dr 0

1.5

Some Special Integrals

Some integrals appear so frequently in engineering mathemat-ics that they deserve special attention. l\1ost of these special

Calculus 1.5 Some Special Integrals

integrals are also called special functions as they have certain varying parameters or integral limits. We only discuss four of the n1ost common integrals here.

1.5.1

Asymptotic Series

Before we discuss any special functions, let us digress first to introduce the asymptotic series and order notations because they will be used to study the behaviours of special functions. Loosely speaking, for two functions

f (

x) and g( x), if

f(x) ---+ K

g(x) ' X---+ XQ, where K is a finite, non-zero limit, we write

f

=

O(g).

(1.57)

(1.58) The big

0

notation means that

f

is asymptotically equivalent to the order of g(x). If the limit is unity or K = 1, we say f(x)

is order of g( x). In this special case, we write

J~g, (1.59)

which is equivalent to f fg ---+ 1 and gf f---+ 1 as x ---+ xo.

Ob-viously, x0 can be any value, including 0 and oo. The notation

~ does not necessarily mean ~ in general, though they might give the same results, especially in the case when x ---+ 0 [for example, sin x ~ x and sin x ~ x if x ---+

0].

When we say

f

is order of 100 (or

f

~ 100), this does not mean

f

~ 100, but it can mean that

f

is between about 50 to 150. The small o notation is used if the limit tends to 0. That is

f

----+ 0, g X---+ XQ, (1.60) or

f

= o(g). (1.61)

If g

>

0,

f

= o(g) is equivalent to

f

<<

g. For example, for Vx E 'R, we have

ex~

1

+

x

+

O(x2)

~

1

+

x

+

~

+

o(x).

Another classical example is the Stirling's asymptotic series for factorials

n

>>

1. (1.62) In fact, it can be expanded into more terms

. ~ n n 1 1 139

nl ' V v ..::;1rn(;) (1

+

12n

+ 288n2 - 51480n3 - ... ).

(1.63)

This is a good example of asymptotic series. For standard power expansions, the error Rk(hk) --+ 0, but for an asymptotic

series, the error of the truncated series Rk decreases and gets smaller con1pared with the leading term [here V21m(nfe)n]. However, Rn does not necessarily tend to zero. In fact,

R2

=

tin·

V21m(nje)n is still very large as

R2

--+ oo if n

>>

1. For

example, for n

=

100, we have n!

=

9.3326 x 10157 , while the leading approximation is V21m(nfe)n = 9.3248 x 10157. The difference between these two values is 7. 77 40 x 10154

, which is still very large, though three orders smaller than the leading approximation.

1.5.2 Gaussian Integrals

The Gaussian integral appears in n1any situations in engineer-ing mathematics and statistics. It can be defined by

!

00

2

/(a) = -oo e-a:x dx. (1.64) In order to evaluate the integral, let us first evaluate /2 . Since the Gaussian integral is a definite integral and must give a constant value, we can change the dummy variable as we wish. We have

1

00

2

100

2

100

2

Calculus 1. 5 Some Special Integrals

=

i: i:

e-o(x2+y2)dxdy. (1.65) Changing into the polar coordinates ( r, 0) and noticing r2

₌

x2

+

y2 and dxdy = rdrdO, we have

roo

[21r

2 I2 =

Jo

dr

Jo

re-ar dO

lo

oo 1 -ar2d( 2) 7r

=

21r -e ar

= -.

o

a a (1.66) Therefore,

( ) 1

00 -ax2d

~

I a

=

e x

= -.

-oo a (1.67)

Since a is a parameter, we can differentiate both sides of this equation with respect to a, and we have

!

00

2 -ox2d 1

~

X e X = - - .

-oo 2a a (1.68)

By differentiating both sides of the Gaussian integral ( equa-tion 1.67) n times with respect to a, and we get the generalized Gaussian integral In I

1

00 2n -ox2 n = X e -oo (1.69) where n

>

0 is an integer.

For a special case when a = ~ and n = 0, the equation v2u"" (1.67) can be rearranged as / 00 00 f(x, a)dx = 1, (1.70)

The function f(x, a) is a zero-mean Gaussian probability func-tion. As a~ 0, f(x) ~ 8(x) where 8(x) is the Dirac 8-function which is defined by

and

/00

00

o(x)dx = 1. (1.72) It has an interesting property that

J

f(x)o(x- (3)dx = !((3), (1.73) where f(x) is a function.

1.5.3 Error Functions

The error function, which appears frequently in heat conduc-tion and diffusion problems, is defined by

2

rx

2

erf(x) =

v'i

Jo e-11 d'f/.

Its complementary error function is defined by 2

1

00 2

erfc(x) = 1- erf(x) = y'i x e-t dt. The error function is an odd function: erf( -x)

Using the results from the Gaussian integral

1

00 2 e -11 _{d'TJ =}

_.;:i,

-oo (1.74) (1.75) = -erf(x). (1.76) together with the basic definition, we have erf(O) 0, and erf( oo) = 1. Both the error function and its complementary function are shown in Figure 1.5.

The error function cannot be easily evaluated in closed form. Using Taylor series for the integrand

_112 - 2 1 4 1 6

e - 1 - 'TJ

+

2"' - 6"'

+ ... ,

(1.77)

and integrating term by term, we have 2 x3 _x5 _x7

Calculus 1.5 Some Special Integrals or 2.5r---~---~---r=========~ 2

-1.5 1 .!So. "t: 0.5 CD 0 -0.5 -1~---1.:.~o _{-5 0} X

Figure 1.5: Error functions.

- erf(x) --- erfc(x) 5 10 2 oo ( -1) n x2n+ 1 erf(x) = -

L

----1- . (1.79)

...fi

n=O 2n

+

1 n.

The integrals of the complementary function are defined by ierfc(x) =

Loo

erfc(TJ)dTJ, (1.80) and

(1.81) Using integration by parts, we can prove the following asymp-totic series

e-x2

erf(x) "-J 1- r-;;' (x----? oo). (1.82)

Xy7r

On the other hand, if we replace x in the error function by

{3x, we have lim -2 1 [1

+

erf(/3x)] ----? H(x),

[3-oo

(1.83)

where H ( x) is a Heaviside function or a unit step function which is defined by

At x = 0, it is discontinuous and it is convention to set H(O) = 1/2. Its relationship with the Dirac 8-function is that

d~~x)

=

8(x). (1.85)

1.5.4 Gamma Functions

The special function is the gamma function which is defined by r(x)

=

fooo

tx-le-tdt

=

fooo

e-t+(x-l)lntdt. (1.86) Using integral by parts, we have

r(x

+

1)

=

fooo

txe-tdt

=

-txe-tl:

+

fooo

xtx-le-tdt

= xr(x). (1.87)

When x

=

1, we have

The variation of r(x) is shown in Figure 1.6.

If X= n is an integer (n E

JV),

then r(n

+

1) = n!. That is to say,

n!

=

r(n

+

1)

=

fooo

enlnt-tdt. (1.89) The integrand f(n, t) = exp[n ln t-t] reaches a tnaximum value at

af

=O

8t ' or t =n. (1.90)

The maximum is !max

=

exp[n ln n-n). Thus, we now can set

t = n

+

r = n(1

+ () so that

r = n( varies around n and ( around 0. For n

>>

1, we have

n! = /_: e{n lnln(l+()J-n(l+<)} dr

Calculus 1.5 Some Special Integrals 20 10 -10 -20 -5 0 X Figure 1.6: Variation of f(x). 5

where we have used ln[n(l + ()] = Inn+ ln(l + (). The inte-gration limits for r = n( (not () are from -oo to oo. Using

(2 (3

ln(l

+ ()

= (-

2" +

"3 - ... ,

(1.92) we have

(1.93) From the Gaussian integral with a= 1/(2n)

1

00 _e-ar 2 _dr

= - =

~

-/2rn,

-oo a (1.94)

we now obtain the Stirling's asymptotic formula

From the basic definition, it can be shown that

r( 1 )

₂

= y7r, r,;;

r(~2)

= ..fi27r. (1.96) The standard gan1ma function can be decomposed into two incomplete functions: the lower incomplete gamma function

/'(0:, x) and the upper incomplete gamnla function r(a, x) so that r(x) = 'Y(O:, x)

+

r(a, x).

The lower incomplete gamma function is defined by (1.97) while the upper incomplete gamma function is defined by

r(a,x) =

1

00

to:-le-tdt. (1.98) Obviously, l(o:,x) ~ r(a) as X~ 00. As r(!)

=

.Jif,

we have

1 1 2 erf(x) =

fi'Y(

2

,x ). (1.99)

Another related function is a beta function

(1.100) From the definition, we know that the beta function is sym-metric, B(x, y) = B(y, x). The beta function is linked to r function by

B( _x,y ) = r(x)r(y) _{r(x +Y).} (1.101)

1.5.5 Bessel Functions

Bessel functions come from the solution of the Bessel's equation (1.102) which arises from heat conduction and diffusion problems as well as wave propagation problems. The solution (see later chapters in this book) can be expressed as Taylor's series, and the Bessel function associated with this equation can be defined by

Calculus 1.5 Some Special Integrals - A.=O -- 1 --- 2 ... -1 -0.5 - 1o!::---=-5 ----1~0=---:-'15' X

Figure 1.7: Bessel functions.

where A is a real paran1eter. These are the Bessel functions of the first kind. It can also be defined by the Bessel integral

1

[27r

J>..(x) =

27r

Jo

cos[ At-x sin t]dt. (1.104) The Bessel functions of the second kind are related to J >.., and can be defined by

y _ J>.. cos(A7r)- J_>..

>.. - sin(A7r) · (1.105)

When A = k is an integer, they have the following properites

The Bessel functions of the first kind are plotted in Figure 1. 7. With these fundamentals of preliminary mathematics, we are now ready to study a wide range of mathematical methods in engineering.

Chapter 2

Vector Analysis

1\tiany quantities such as force, velocity, and deformation in en-gineering and sciences are vectors which have both a n1agnitude and a direction. The manipulation of vectors is often associated with matrices. In this chapter, we will introduce the basics of vectors and vector analysis.

2.1

Vectors

A vector x is a set of ordered numbers x = (xt, x2, ... , Xn), where its components x_{1 ,} x_{2, ... , Xn are real numbers. All these}

vectors form a n-dimensional vector space

vn.

To add two vectors x = (xb x2, ... , Xn) andy = (Yt, Y2, ... , Yn), we simply add their corresponding components,

Z =X+ Y = (xt + Yt,X2 + Y2, ... ,Xn + Yn), (2.1) and the sum is also a vector. This follows the vector addition parallelogram as shown in Fig 2.1

The addition of vectors has comn1utability (u

+

v = v

+

u) and associativity [(a+ b) + c = a+ (b +c)]. Zero vector 0 is a special vector that all its components are zeros. The multiplication of a vector x with a scalar or constant a: is carried

out by the multiplication of each component,

ay = (ay1, oy2, ... , ayn)· (2.2) Thus, -y

= (

-y~, -y2, ... , -Yn)· In addition, (a{3)y

=

a(f3y) and

(a+

f3)y = ay

+

{3y.

4 ...•...•... ~···

')/

:" .· ...

.

·

.: : : // .· .· : :·

Figure 2.1: Vector addition.

Two nonzero vectors a and b are said to be linearly inde-pendent if aa

+

{3b

=

0 implies that a= {3

=

0. If a, {3 are not all zeros, then these two vectors are linearly dependent. Two linearly dependent vectors are parallel ( a//b) to each other. Three linearly dependent vectors a, b, c are in the same plane.

2.1.1 Dot Product and Norm

The dot product or inner product of two vectors x and y is defined as

n

X· Y

=

XtYl

+

X2Y2

+ ··· +

XnYn

=

L

XiYi, (2.3) i=l

which is a real number. The length or norm of a vector x is the root of the dot product of the vector itself,

Vector Analysis 2.1 Vectors

When

llxll

= 1, then it is a unit vector. It is straightforward to check that the dot product has the following properties:

X·Y = y ·X, X· (y+z) =x·y+x·z, (2.5) and

(a:x) · (f3y) = (a:f3)x · y, (2.6) where a:, {3 are constants.

If() _{is the angle between two vectors a and b, then the dot}

product can also be written

a· b

=

llall llbll

cos(8), 0 ::; () ::; 7r. _(2.7) If the dot product of these two vectors is zero or cos(8)

=

0 (i.e., () = 1r /2), then we say that these two vectors are orthogonal.

Rearranging equation (2. 7), we obtain a formula to calcu-late the angle () between two vectors

a·b

cos(()) =

II all

lib II

(2.8) Since cos(8) ::; 1, then we get the useful Cauchy-Schwartz in-equality:

(2.9) Any vector a in a n-dimensional vector space

vn

can be written as a combination of a set of n independent basis vectors or orthogonal spanning vectors e1, e2, ... , en, so that

n a= a:1e1

+

a:2e2 + ... + O:nen =

L

O:iei,

i=l

(2.10)

where the coefficients/scalars 0:1,0:2, ... , O:n are the components of a relative to the basis e1, e2 ... , en. The most common basis vectors are the orthogonal unit vectors. In a three-dimensional case, they are i = (1, 0, 0), j = (0, 1, 0, k = (0, 0, 1) for three x-, y-, z-axis, and thus x = x1i + x2.i

+

x3k. The three unit vectors satisfy i · j = j · k = k · i = 0.

2.1.2 Cross Product

The dot product of two vectors is a scalar or a number. On the other hand, the cross product or outer product of two vectors is a new vector

c=axb

which is usually written as

i j k a X b = Xt X2 X3

Yt Y2 Y3

=

I :: :: li +I :: :: lj +I :: :: lk.

(

2.12) The angle between a and b can also be expressed as

. lla x bll

sin f) = llallllbll. (2.13)

In fact, the norm lla x bll is the area of the parallelogram formed by a and b. The vector c =ax b is perpendicular to both a and b, following a right-hand rule. It is straightforward to check that the cross product has the following properties:

xxy = -yxx, (x

+

y)xz = xxz

+

yxz, (2.14) and

(ax)x (,By) = (a,B)xxy. (2.15) A very special case is axa = 0. For unit vectors, we have

ixj = k, jxk = i, kxi = j. (2.16)

0 Example 2.1: For two 3-D vectors a = (1, 1, 0) and b =

(2, -1, 0), their dot product is

Vector Analysis 2.1 Vectors

As their moduli are

we can calculate the angle(} between the two vectors. We have

a·b

1

cosO=

llallllbll

=

J2J5'

or

(} _{= cos} - l _1'17\ 1 _{~ 71.56 .} 0 vlO

Their cross product is

V =a X b = (1 X 0-0 X (-1),0 X 1-1 X 0,1 X (-1)- 2 X 1)

=

(0,0, -3),

which is a vector pointing in the negative z-axis direction. The vector

v is perpendicular to both a and b because

a · V = 1 X 0

+

1 X 0

+

0 X ( -3) = 0, and

b · V = 2 X 0

+ (

-1) X 0

+

0 X ( -3) = 0.

0

2.1.3 Vector Triple

For two vectors, their product can be either a scalar (dot prod-uct) or a vector (cross prodprod-uct). Similarly, the product of triple vectors a, b, c can be either a scalar

ax ay a:: a· (b x c)= bx by b::

'

(2.17) Cy Cz or a vector ax (b x c)= (a· c)b- (a· b)c. (2.18)

As the dot product of two vectors is the area of a parallel-ogram, the scalar triple product is the volume of the paral-lelepiped formed by the three vectors. From the definitions, it is straightforward to prove that

a· (b x c)= b · (c x a)= c ·(ax b)= -a· (c x b), (2.19) a X (b X c) #(a X b) XC, (2.20) and

(ax b)· (c x d) =(a· c)(b ·d)- (a· d)(b ·c). (2.21)

2.2 Vector Algebra

2.2.1 Differentiation of Vectors

The differentiation of a vector is carried out over each compo-nent and treating each con1pocompo-nent as the usual differentiation of a scalar. Thus, for a position vector

P(t) = x(t)i

+

y(t)j

+

z(t)k, (2.22) we can write its velocity as

v

=

~

=

±(t)i + y(t)j

+

z(t)k, (2.23) and acceleration as

a=

~t~

= X(t)l

+ ij(t)j +

Z(t)k, (2.24) where () = d()jdt. Conversely, the integral of vis

P =

J

vdt+c,

where cis a vector constant.

Vector Analysis 2.2 Vector Algebra

From the basic definition of differentiation, it is easy to check that the differentiation of vectors has the following prop-erties: d(a:a) da d(a ·b) da b db ~=0_{dt~ dt} = dt ·

+a·dt'

(2.26) and d(axb) _ da b db dt - dt x

+

ax dt · (2.27)

2.2.2 Kinematics

As an application of vector algebra, let us study the motion along a curved path. In mechanics~ there are three coordinate systems which can be used to describe the motion uniquely. The first one is the Cartesian coordinates (x, y) with two unit vectors i (along positive x-axis) and j (along positive y-axis), and the second one is the polar coordinates (r, 0) with two unit vectors er and ee as shown in Figure 2.2.

y

Figure 2.2: Polar coordinates, their unit vectors and their re-lationship with Cartesian coordinates.

The position vector r

=

x(t)i

+

y(t)j at point P at any instance t in the Cartesian coordinates can be expressed as ( r, 0). The velocity vector is

and the acceleration is

.2 .. .

a=

v

= (f-rO )er

+

(rO

+

2rO)eo. (2.29) The third coordinate system is the intrinsic coordinate sys-tem (s, 'l/J) where sis the arc length from a reference point (say, point 0) and 'l/J is the angle of the tangent at the point P (see Figure 2.3). The two unit vectors for this systems are et along the tangent direction and en which is the unit normal of the curve.

Figure 2.3: Intrinsic coordinates and their unit vectors. In the intrinsic coordinates, the position is uniquely deter-mined by (s,

1/J),

and the velocity is always along the tangent. Naturally, the velocity is simply

(2.30) The acceleration becomes

.. 82

a= set+ -en,

p (2.31)

where pis the radius of the curvature at point P.

For the circular n1otion such as a moving bicycle wheel as shown in Figure 2.4, the three coordinate systems are intercon-nected. In a rotating reference frame with an angular velocity

Vector Analysis 2.2 Vector Algebra

y

Figure 2.4: Three coordinate systems for a wheel in circular motion.

w

=

Ok

_{where k point to the z-axis, the velocity and} accelera-tion at any point (say) P can be calculated using another fixed point A on the rotating body (or wheel). The velocity is

drl

V p

=

VA

+ dt

A

+

W X r, (2.32)

and the acceleration is d2

rl

dw

ap

=

aA

+ dt2

A+

-;Jf

X r

+

acor

+

acent, (2.33)

where

drl

acor = 2w X dt A' (2.34) is the Coriolis acceleration, and

acent = W X ( W X r), (2.35) is the centripetal acceleration. It is worth noting that the ve-locity v A and acceleration aA is the velocity and acceleration

in a non-rotating frame or an inertia frame.

In addition, the differentiation of the unit vectors are con-nected by

and

(2.37) In the intrinsic coordinates, we have s = R</> where R =constant is the radius of the wheel in circular motion. Thus, 8 = R~. The velocity for this circular motion is simply

(2.38) Differentiating it with respect to time and using the relation-ships of unit vectors, we have

• .. '2

a = v = R</>et

+

R</> en, (2.39) where the unit vectors are

et =cos </>i +sin </>j, en = -sin </>i +cos </>j. (2.40)

0 Example 2.2: A car is travelling rapidly along a curved path with a speed of 30 mjs at a given instance. The car is fitted with

an accelerometer and it shows that the car is accelerating along the

curved path at 2 m/~. The accelerometer also indicates that the component of the acceleration perpendicular to the travelling direc-tion is 5 mjs2. lVhat is the direcdirec-tion of the total acceleradirec-tion at this instance? lVhat is the radius of the curvature? Suppose the car has

a height of 2 meters and a width of 1.6 meters, is there any danger

of toppling over?

Let () be the angle between the acceleration vector and the veloc-ity vector, and let a be the magnitude of the total acceleration. In the intrinsic coordinates, the velocity is v =set= 30et. The accelera.tion is given by

·2

a= set+ ~en= a(cos8et

+

sin8en)· p

Therefore, we have

s

2 ₃₀2

- = -

=

a sin()

=

5,

Vector Analysis 2.2 Vector Algebra

or the instantaneous radius of curvature is p = 302_{/5 = 180m. \-Ve}

know that the magnitude of the acceleration is a =

v2

2

+

_{52 =}

J29.

The angle is

5

(} = tan-1

'2

~ 68.20°.

In addition, we can assume that the centre of gravity is approx-imately at its geometrical centre. Thus, the centre is 1m above the

road surface and 0.8m from the edges of the outer wheels. If we take the moment about the axis through the two contact points of the outer wheels, we have the total moment

v2

1 X 1\t[--0.8Mg ~ -2.8A1

<

0,

p

where J\tl is the mass of the car. There is no danger of toppling over. However, if the car speeds up to v = 42 m/s (about 95 miles per hour), there is a danger of toppling over when the moment of the weight is just balanced by the moment of the centripetal force. 0

2.2.3 Line Integral

I

ldy I

________

_.

dx ds = Jdx'l

+

dy'l

Figure 2.5: Arc length along a curve.

An important class of integrals in this context is the line integral which integrates along a curve r(x, y, z)

=

xi+ yj + zk. For example, in order to calculate the arc length L of curve r

as shown in Figure 2.5, we have to use the line integral.

L

=

[s

ds

=

[s

J

dx2

+

dy2

=

1x

}~

~0

xo (2.41)

0 Example 2.9: The arc length of the parabola y(x) = tx2 _from

x = 0 to x = 1 is given by

L =

1'

v't

+

y'2dx =

1'

v't

+

x2dx

=

~[x~

+

ln(x

+

~>{

1

=

₂[V2 -ln(V2 -1)] ~ 1.14779.

2.2.4 Three Basic Operators

Three important operators commonly used in vector analysis, especially in fluid dynamics, are the gradient operator (grad or V), the divergence operator ( div or V ·) and the curl operator (curl or Vx).

Sometimes, it is useful to calculate the directional derivative of a function ¢ at the point ( x, y, z) in the direction of n

a¢

a¢

a¢

8¢

an

=

n · V¢

=

ax

cos(

a)+

ay

cos(f3)

+

8z

cos( I), (2.42)

where n

=

(cosa,cos{3,cos1) is a unit vector and o.,f3,1 are the directional angles. Generally speaking, the gradient of any scalar function¢ of

x,

y,

z

can be written in a sin1ilar way,

(2.43) This is equivalent to applying the del operator V to the scalar function¢

r7

8.

8.

8k

v =

-1+-J+-.

Vector Analysis 2.2 Vector Algebra

The direction of the gradient operator on a scalar field gives a vector field. The gradient operator has the following properties: \l(o:'l/J+/3¢) = o:\11/J+/3\1¢, \7(1/J¢)

=

1/J\14>+¢\11/J, (2.45) where a, (3 are constants and 1/J, 4> are scalar functions.

For a vector field

u(x, y, z) = ut(x, y, z)i

+

u2(x, y, z)j

+

u3(x, y, z)k, (2.46) the application of the operator \1 can lead to either a scalar field or a vector field, depending on how the del operator applies to the vector field. The divergence of a vector field is the dot product of the del operator \1 and u

dlVU= . t"'7 v

· U = - + - + -

8ut 8u2 8u3

8x

oy

oz'

(2.47)

and the curl of u is the cross product of the del operator and the vector field u

i j k

curl u = \1 x u = _(JX 1:) _(JY 1:) _IE {) (2.48)

Ut u2 u3

It is straightforward to verify the following useful identities associated with the \1 operator:

\1 · \1 X U

=

0, (2.49) \1 X \11/J = 0, (2.50) \1 X (1/Ju) = 1/J\1 Xu+ (\11/J) XU, (2.51) 'V· (1/Ju) = 1/J\1· u + (\11/J) · u, (2.52) \1 x (\1 x u) = \1(\1 · u) - \72u. (2.53) One of the most common operators in engineering and sci-ence is the Laplacian operator is

2

8

2

w

8

2

w

8

2

w

for Laplace's equation

(2.55) In engineering mathematics, it is sometimes necessary to ex-press the Laplace equation in other coordinates. In cylindrical polar coordinates (r, </>, z), we have

\7. u

=

~ 8(rur)

+

~ 8ucp

+

8u::

r 8r r 8</> 8z · (2.56)

The Laplace equation becomes

(2.57) In spherical polar coordinates (r, 8, ¢), we have

\7. u = 1 82(r2ur)

+

1 8(sin8ue)

+

_1_8uct> (_2.58)

r2 8r2 r sin 8 88 r sin 8 8<1> ·

The Laplace equation can be written as

(2.59)

2.2.5 Some Important Theorems

The Green theorem is an important theorem, especially in fluid dynamics and the finite element analysis. For a vector field Q

=

ui

+

vj in a 2-D region

n

with the boundary

r

and the unit outer normal nand unit tangent t. The theorems connect the integrals of divergence and curl with other integrals. Gauss's theorem states:

Jjfn(v.

Q)dn =

Jfs

Q. ndS, (2.60) which connects the volume integral to the surface integral.

Vector Analysis 2.3 Applications

Another important theorem is Stokes's theorem:

Jfs(v

x Q). kdS

=

£

Q. tdr

=

£

Q. dr, (2.61) which connects the surface integral to the corresponding line integral.

In our simple 2-D case, this becomes

f

(udx

+

vdy) =

J

[ [

ln (

8v

8u

0x - {)y)dxdy. (2.62) For any scalar functions

'l/J

and ¢, the useful Green's first identity can be written as

J

'l/JV</>dr

= { (,PV2¢

+

V'l/1 ·

V¢)d0., (2.63)

!on

ln

where dO. = dxdydz. By using this identity twice, we get Green's second identity

2.3 Applications

In order to show the wide applications of vector analysis, let us apply them to study the mechanical and flow problems.

2.3.1 Conservation of Mass

The mass conservation in flow mechanics can be expressed in either integral form (weak form) or differential form (strong form). For any enclosed volume

n,

the total mass which leaves or enters the surface S is

fspu · dA,

where p(x, y, z, t) and u(x, y, z, t) are the density and the ve-locity of the fluid, respectively. The rate of change of mass in

n

is

The n1ass conservation requires that the rate of loss of mass through the surface S is balanced by the rate of change in

n.

Therefore, we have

Using Gauss's theorem for the surface integral, we have

In

V · (pu)dV +

!

In

pdV

=

0.

Interchange of the integration and differentiation in the second term, we have

1

8p

[- +

V · (pu)]dV = 0. n

8t

This is the integral form or weak form of the conservation of mass. This is true for any volume at any instance, and subse-quently the only way that this is true for all possible choice of

n

is

ap

8t+V·(pu)=0,

which is the differential form or strong form of mass conser-vation. The integral form is more useful in numerical meth-ods such as finite volume methmeth-ods and finite element methmeth-ods, while the differential form is more natural for mathmnatical analysis.

2.3.2 Saturn's Rings

We all know that Saturn's ring system ranks among the most spectacular phenomena in the solar system. The ring systen1 has a diameter of 270,000 km, yet its thickness does not ex-ceed 100 meters. The sizes of particles in the rings vary from centimeters to several meters, and this size distribution is con-sistent with the distribution caused by repeated collision. The ring system has very complicated structures. One natural ques-tion is why the formed structure is a ring system, why not a